Metadynamics method

Stirling, A. Iannuzzi, M. Laio, A. Parrinello, M., Azulene-to-naphthalene rearrangement the Car-Parrinello metadynamics method explores various reaction mechanisms, Eur. J. Chem. Phys. Phys. Chem. 2004, 5, 1558-1568... 

If the metadynamics method is applied to the simulation of chemical reactions in conjunction with Car-Parrinello molecular dynamics [36,40,49], the history dependent potential has to force the system to cross barriers of several tenths of kcal/mol in a very short time, usually a few picoseconds. This implies that a lot of energy is injected in the degrees of freedom associated with the collective variables. This might lead to a significant dishomogeneity in the temperature distribution of the system, and possibly to instabilities in the dynamics. 

The metadynamics method was introduced in 2002 by Laio and Parrinello as an elegant extension of adaptive bias potential methods [65]. The authors used a coarse-grained non-Markovian dynamics in the space defined by a few collective coordinates s,. With the aid of a history-dependent potential term the minima of the FES were filled in time, allowing the efficient exploration and accurate determination of the FES as a function of the collective coordinates. Laio and Parrinello demonstrated the appUcabUity of this approach in the case of the dissociation of a sodium chloride molecule in water and in the study of the conformational changes of a dialanine in solution [65]. 

Abstract This chapter offers an overview of recent applications of the metadynamics method to the study of nucleation and related phenomena. In the first section, the classical nucleation theory and the metadynamics method are introduced. The second section is devoted to applications, including computational studies of the surface tension, which affects the size and energy of criticial nuclei, and investigation of crystal nucleation fi om the amorphous and supercooled liquid state. 

In this introductory subsection, we discuss the classical theory of homogeneous nucleation and of crystal growth and the metadynamics method, including a mention of recent advances aiming at increasing the efficiency and the flexibility of this... 

Biarnes, X., Bongarzone, S., Vargiu, A. V., Carloni, R, Ruggerone, P. (2011). Molecular motions in drug design The coming age of the metadynamics method. Journal of Computer-Aided Molecular Design, 25, 395. 

As the walker leaves free energy minima most easily through saddle points in the free energy landscape, the metadynamics method is also capable of discovering new pathways between stable states and it can help to identify the respective mechanisms. Of course, the method is successful only if the space of the collective variables includes the degrees of freedom, which properly characterize the... 

In this chapter, we focus on the method of constraints and on ABF. Generalized coordinates are first described and some background material is provided to introduce the different free energy techniques properly. The central formula for practical calculations of the derivative of the free energy is given. Then the method of constraints and ABF are presented. A newly derived formula, which is simpler to implement in a molecular dynamics code, is given. A discussion of some alternative approaches (steered force molecular dynamics [35-37] and metadynamics [30-34]) is provided. Numerical examples illustrate some of the applications of these techniques. We finish with a discussion of parameterized Hamiltonian functions in the context of alchemical transformations. 

Fig. 4.16. The A dynamics method for alchemical transformations was developed by Guo and Brooks [57] for rapid screening of binding affinities. In this approach the parameter A is a dynamic variable. Techniques like ABF or metadynamics [34] can be used to accelerate this type of calculation. A dynamics was used by Guo [57] to study the binding of benzamidine to trypsin. One simulation is sufficient to gather data on several benzamidine derivatives. Substitutions were made at the para position C5 (H, NH2, CH3 and Cl). The hydrogen atoms are not shown for clarity...

Another apparent difference between the various free energy methods lies in the treatment of order parameters. In the original formulation of a number of methods, order parameters were dynamical variables - i.e., variables that can be expressed in terms of the Cartesian coordinates of the particles - whereas in others, they were parameters in the Hamiltonian. This implies a different treatment of the order parameter in the equations of motion. If one, however, applies the formalism of metadynamics, or extended dynamics, in which any parameter can be treated as a dynamical variable, most conceptual differences between these two cases vanish. 

Metadynamics is first used to reconstruct the free energy as a function of many simple coordinates at the same time. The filling rate w/tq and the Gaussian width Ss have to be chosen in order to provide the possibility to explore completely the free energy surface in the available computational time. This requirement, in high dimensionality, necessarily implies a large error, especially if the method is used in conjunction with Car-Parrinello molecular dynamics [49]. 

F. Wang and D. Landau (2001) Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett. 86, p. 2050 G. Bussi, A. Laio, and M. Parrinello (2006) Equihbrium free energies horn non-equilibrium metadynamics. Phys. Rev. Lett. 96, p. 090601 R. Martonak, A. Laio, and M. Parrinello (2003) Predicting crystal structures The parrineUo-rahman method revisited. Phys. Rev. Lett. 90, p. 75503... 

Laio, A., and Gervasio, F. L. (2008). Metadynamics A method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science. Reports on Progress in Physics 71,12, p. 126601. 

Barducci, A., Bussi, G., Parrinello, M. Well-tempered metadynamics a smoothly converging and mnable free-energy method. Phys. Rev. Lett 100, 020603 (2008)... 

Metadynamics is a method based on MD simulations that allows thorough sampling of a predefined multidimensional configurational space and provides, at the same time, the direct reconstruction of the explored free energy surface (FES) [65-71]. 

Next, difficulties encountered in ab initio molecular dynamics simulations were discussed. Topics covered were massive parallelization to address computer time problems, basis set considerations, density functionals and van der Waals interactions, relativistic corrections, and new integration schemes. Several simulation techniques used to gain chemical insight were summarized. Enhanced sampling methods, metadynamics and other methods to explore free energy surfaces, reaction pathways and transition states were covered. Simulation of spectra (IR, NMR, EXAFS) from ab initio molecular dynamics simulations was the subject of the remaining paragraphs. 

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